The four-lectures course is addressed to practitioners of standard Finite Element (FE) methods familiar with basic variational formulations, the (Bubnov–)Galerkin method and the standard technology of FEs. The class combines a short introduction to the “Discontinuous Petrov–Galerkin (DPG) Method with Optimal Test Functions” with a crash course on the energy spaces forming the exact sequence and the corresponding conforming FE discretizations. We will introduce the participants to hp3D—a 3D MPI/OpenMP code supporting hp-discretizations of the exact-sequence elements on hybrid (tets, cubes, prisms, pyramids) meshes and demonstrate how to implement the DPG method in such a framework. On the application side, we will focus on wave propagation problems: time-harmonic acoustics, Maxwell’s equations, and elastodynamics.
Syllabus:
- Lecture 1
- Examples of variational formulations with symmetric and non-symmetric functional setting; brief introduction to energy spaces.
- A crash course on H1, H(curl), H(div), and L2-conforming finite elements.
- Lecture 2
- Introduction to the hp3D code.
- Examples of applications of the Bubnov–Galerkin method.
- Lecture 3
- A crash course on the DPG method.
- Implementation of DPG in the hp3D code.
- Lecture 4
- MPI/OpenMP parallel computation with the hp3D code.
- Examples of applications of the DPG method.
Course Material:
- L. Demkowicz. Lecture Notes on Energy Spaces. Technical Report 13, ICES, 2018. https://www.oden.utexas.edu/media/reports/2018/1813.pdf
- L. Demkowicz. Lecture Notes on Mathematical Theory of Finite Elements. Technical Report 11, Oden Institute, June 2020. https://www.oden.utexas.edu/media/reports/2020/2011.pdf
- S. Henneking and L. Demkowicz. Computing with hp Finite Elements III. Parallel hp Code. 2022. In preparation, available upon request.
Software:
- hp3D open-source library, GitHub: https://github.com/Oden-EAG/hp3d
- hp3D user manual, arXiv:2207.12211 (2022): https://arxiv.org/pdf/2207.12211.pdf
